914 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 7, JULY 2002 w-Channel Linear Phase Perfect Reconstruction Filter Bank With Rational Coefficients Trac D. Tran, Member, IEEE Abstract—This paper introduces a general class of -channel transform (DCT) which can be viewed as an irrational-coef- linear phase perfect reconstruction filter banks (FBs) with rational ficient 8-channel 8-tap linear phase orthogonal FB [13], [14]. coefficients. A subset of the presented solutions has dyadic coeffi- The future image compression standard JPEG2000 also utilizes cients, leading to multiplierless implementations suitable for low- power mobile computing. All of these FBs are constructed from several biorthogonal integer-coefficient wavelet pairs for fast a lattice structure that is VLSI-friendly, employs the minimum coding/decoding and for lossless compression [15], [16]. number of delay elements, and robustly enforces both linear phase This paper introduces a large family of FIR linear phase and perfect reconstruction property. The lattice coefficients are pa- perfect reconstruction FBs with rational coefficients and rameterized as a series of zero-order lifting steps, providing fast, good energy compaction property. Integer implementations efficient, in-place computation of the subband coefficients. Despite the tight rational or integer constraint, image coding experiments can be easily found via a common scaling factor. A tighter show that these novel FBs are very competitive with current pop- constraint yields solutions with dyadic coefficients, which lead ular transforms such as the 8 8 discrete cosine transform and the to efficient multiplierless implementations. Our focus is on the wavelet transform with 9/7-tap biorthogonal irrational-coefficient construction of the polyphase matrices as cascades of low-order filters. modular components. Desirable properties such as symmetry, Index Terms—Compression, dyadic coefficients, linear phase FIR, and perfect invertibility, are propagated by imposing them filter bank, multiplierless, rational coefficients. structurally onto each cascaded module. I. INTRODUCTION A. Outline ULTIRATE filter banks (FBs) have found tremendous The outline of the paper is as follows. In Section II, we offer M applications in the analysis, processing, and efficient a review of important background materials, concepts, motiva- representation of digital signals [1]–[4]. Signal representations tions, and previous related works in multirate FB design using by subband samples are usually more compact, more efficient, lattice and ladder structures. The next section introduces a gen- yet as informative as the time-domain counterparts. Taking eral parameterization of polyphase matrices based on lifting advantage of the normally sparse subband sample matrix, we steps (also known as ladder structures) and the subset of so- can often obtain significant data compression. lutions that allows the construction of -band rational- and One particular class of FBs that have attracted a lot of dyadic-coefficient FBs. Parts of Sections II and III are meant recent interests is FBs with integer coefficients [5]–[12]. to serve as tutorial materials. Design issues and various design First of all, integer-coefficient FBs eliminate the truncation examples are presented and discussed in Section IV. The suc- error in finite-precision implementations. More importantly, cessful application of the newly found family of FBs in image integer-arithmetic implementations in hardware are faster, coding is illustrated in Section V. Finally, Section VI ends the require less chip area, and consume less power. Thirdly, many paper with a brief summary. integer-coefficient FBs also have very fast multiplierless imple- mentation with simple shift-and-add operations only. Hence, B. Notations integer FBs are desirable in applications with high data rates Let , , and denote the sets of real numbers, rational as well as in portable computing and wireless communication numbers, and integers, respectively. Also, let denote the set applications. Integer FBs or integer approximations are already of dyadic rationals, i.e., all rational numbers that can be rep- popular in practice. For example, current international image resented in the form of 2 where . Bold-faced and video compression standards JPEG and MPEG employ lower case characters are used to denote vectors while bold- several integer approximations of the 8-point discrete cosine faced upper case characters are used to denote matrices. , , , and denote, respectively, the transpose, the in- Manuscript received April 23, 2001; revised December 21, 2001. This work verse, the determinant, and the th th element of the matrix . has been supported in part by the National Science Foundation under Grant When the size of a matrix or vector is not clear from context, CCR-0093262 and in part by FastVDO Inc. This paper was presented in part capital subscripts will be included. The notation at the SPIE Wavelets Applications in Signal and Image Processing Conference, Denver, CO, July 1999. This paper was recommended by Associate Editor P. P. or indicates that every element of the matrix Vaidyanathan. is either rational or dyadic, i.e., or . The author is with the Department of Electrical and Computer Engi- Several special matrices with reserved symbols are: the neering, The Johns Hopkins University, Baltimore, MD 21218 USA (e-mail: [email protected]). polyphase matrix of the analysis bank , the polyphase Publisher Item Identifier 10.1109/TCSI.2002.800467. matrix of the synthesis bank , the identity matrix , the 1057-7122/02$17.00 © 2002 IEEE TRAN: -CHANNEL LINEAR PHASE PERFECT RECONSTRUCTION FILTERBANK 915 Fig. 1. w-channel uniform FB. (a) Conventional representation. (b) Polyphase representation. reversal or anti-diagonal matrix , the null matrix , a permuta- synthesis filters have linear phase (their impulse responses are tion matrix , and the diagonal matrix . and are usually either symmetric or antisymmetric if the filters have real coef- reserved for the number of channels and the filter length. In this ficients). Besides the elimination of the phase distortion, linear paper, we only consider the class of FBs whose filters all have phase systems allow us to use simple symmetric extension the same length .An -channel -tap FB methods to accurately handle the boundaries of finite-length is sometimes denoted as an transform. The symbols signals. Furthermore, the linear phase property can be ex- , , and , stand for the th ploited, leading to faster and more efficient FB implementation. analysis filter’s impulse response, its associated -transform, If all filters have the same length , it has been well and its Fourier transform. Similarly, the th synthesis filter established that the polyphase matrices satisfy the following is denoted by , ,or . For abbreviations, symmetric property [2] [17] we often use LP, PR, and FB to denote linear phase, perfect reconstruction, and FB. analysis (1) synthesis (2) II. BACKGROUND AND MOTIVATION where is the diagonal matrix with entries being 1or 1 A. FB Fundamentals depending on the corresponding filter being symmetric or anti- symmetric. In this paper, we shall limit the discussions on discrete-time maximally decimated -channel uniform FBs as depicted in The highly complex problem of designing an -band Fig. 1. At the analysis stage, the input signal is passed FIR linear phase perfect reconstruction FB can be reduced to choosing appropriate polynomial matrices and through a bank of analysis filters , each of which pre- such that both FIR invertibility and linear phase condition serves a frequency band. The overall sampling rate is preserved by the -fold downsamplers. At the synthesis stage, the sub- are satisfied. Additional constraints that yield rational- or bands are combined by a set of upsamplers and synthesis dyadic-coefficient filters will be imposed on top. Knowing and , we can easily find the filters , and filters to form the reconstructed signal . vice versa. This is the approach that will be taken throughout The FB in Fig. 1(a) can also be represented in terms of its this paper. polyphase matrices as shown in Fig. 1(b). Here, is the anal- ysis bank’s polyphase matrix and is the synthesis bank’s polyphase matrix. Note that both and are B. The Lattice Structure matrices whose elements are polynomials in [1]. The delay The lattice structure is one of the most effective and elegant chain and the -fold downsamplers act as a serial-to-parallel tools in -channel FB design and implementation. This ap- converter. Similarly, in the synthesis bank, the -fold upsam- proach bases on various factorizations of the polyphase matrices plers and the delay chain act as a parallel-to-serial converter. and [17]–[23]. From another viewpoint, in the lat- The polyphase representation usually leads to faster and more tice approach, the polyphase matrix or is constructed efficient implementations. It is also very helpful in the FB design from a cascade of modular low-ordered structures that propagate process. If is invertible with monomial determinant, i.e., certain desirable properties (such as linear phase and perfect re- , one can obtain the output as a pure construction). The lattice structure has been known to offer a delayed version of the input , , with FIR fast FB implementation with the minimal number of delay el- synthesis filters by simply choosing [1]–[3]. ements, and it can retain all desirable properties regardless of We call this class of polyphase matrices FIR invertible. The re- lattice coefficient quantization [1]–[4]. sulting FIR FB is said to be biorthogonal or to have perfect re- The general -band linear-phase lattice structure is pre- construction. In the case when , sented below [21]. The polyphase matrix can be factored the FB is called paraunitary. as In numerous practical applications, especially in image and video processing, it is often desired that all analysis and (3) 916 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 7, JULY 2002 Fig. 2. General lattice structure for w-channel LPPRFB with filter length v a uw (drawn for w aV). (a) (b) Fig. 3.